Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 6; Problem 6.4.
It’s useful to do a tensor calculation using actual numbers, just to see how they work. Suppose we have a vector with components
where the units are metres, and a second vector given by
where the units are .
Since the units are different (even using relativistic units where length = time, since vector has units of 1/time), we cannot add these vectors. However, we can multiply them to get a rank-2 tensor:
The first row consists of multiplied by each element of in turn, and so on (the indices and each run for 0 to 3). The units of each element of are .
The question in Moore asks if this product is commutative, which isn’t really the right thing to ask, since obviously . However, the product is not symmetric, in the sense that . That is, it matters which vector is used to determine the row index and which the column index.
The trace of is given by
This is just the scalar product . If we use the special relativity metric , then
We could also have obtained this result from the matrix directly by calculating (the other two diagonal elements are zero).